Math 5334: Differential Geometry
Course Description: This course covers both the
differential and the geometric parts of differential geometry. The differential part includes manifolds, the
tangent bundle of a manifold, differential forms, tensors, and Riemannian
metrics. All of this is used in
formulating the geometric part, which includes curvature and torsion of space
curves, Gaussian curvature of surfaces, the Gauss-Bonnet Theorem, covariant
derivatives, parallel transport, and the Riemann curvature tensor on manifolds
of higher dimension.
List of Topics:
1. Manifolds
Examples of manifolds
Coordinate charts
Transition functions between coordinate
charts
Atlases on manifolds
Differentiability of functions between
manifolds
The Jacobian
The chain rule
2. The
Tangent Bundle
Description of vectors as linear operators
Transformations of vectors under changes of
coordinates
Vector fields
3. The
Cotangent Bundle
Differential 1-forms
Transformations of differential forms under a change of coordinates
4. Tensors
Covariant and contravariant
tensors
Raising and lowering of indices (via a
Riemannian metric)
5. Inner
Products
The matrix of an inner product with respect
to a basis
Riemannian metrics on manifolds
6. Curves in
the Plane and in Space
Curvature and torsion of plane curves and
space curves
The Serret-Frenet
formulas
7. Curvature
of Surfaces in Space
Gaussian curvature
The first and second fundamental forms
The Theorema Egregium
8.
Differential Geometry on Higher-dimensional Manifolds
Riemann curvature tensor (its definition
and meaning)
Torsion tensor (its definition and meaning)
Covariant derivatives and the Levi-Civita connection
Parallel transport
9. The
Gauss-Bonnet Theorem
References:
1. Michael Spivak, A Comprehensive
Introduction to Differential Geometry, Volumes I and II (and Vol. III for
the Gauss-Bonnet Theorem)
2. Manfredo Do Carmo, Differential Geometry of Curves and
Surfaces
3. Michael Spivak, Calculus on Manifolds
4. M.
Nakahara, Geometry and Physics